Existence and uniqueness analysis

\[\begin {aligned} x^{\prime }&=\left (x-1\right )^{2}\\ x \left (0\right ) &= 1\\ \end {aligned}\]

This is non linear ﬁrst order ODE. In canonical form it is written as

\begin{align*} x^{\prime } &= f(t,x)\\ &= \left (x -1\right )^{2} \end{align*}

The \(x\) domain of \(f(t,x)\) when \(t=0\) is

\begin{align*} \{-\infty <x <\infty \} \end{align*}

And the point \(x_0 = 1\) is inside this domain. Now we will look at the continuity of

\begin{align*} \frac {\partial f}{\partial x} &= \frac {\partial }{\partial x}\left (\left (x -1\right )^{2}\right ) \\ &= 2 x -2 \end{align*}

The \(x\) domain of \(\frac {\partial f}{\partial x}\) when \(t=0\) is

\begin{align*} \{-\infty <x <\infty \} \end{align*}

And the point \(x_0 = 1\) is inside this domain. Therefore solution exists and is unique.